Computer Vision Exercise Session 1 Institute of Visual Computing - - PowerPoint PPT Presentation

Institute of Visual Computing

Computer Vision

Exercise Session 1

Institute of Visual Computing

Organization

• Teaching assistant
• Bastien Jacquet
• CAB G81.2
• bastien.jacquet@inf.ethz.ch
• Federico Camposeco
• CNB D102.2
• fede@inf.ethz.ch
• Lecture webpage
• http://www.cvg.ethz.ch/teaching/compvis/index.php

Institute of Visual Computing

Organization

• Assignments will be part of the final grade
• Assignment every week (or every two weeks the second

part)

• 1 or 2 weeks time to solve the assignment
• Hand-in Thursday 13:00 (sharp!)
• Bonus points can be used to compensate for missing

points

• MATLAB Download (www.stud-ides.ethz.ch)

Institute of Visual Computing

Literature

• Computer Vision: Algorithms and Application

by Richard Szeliski, available online ( http://szeliski.org/Book/ )

• Multiple View Geometry

by Richard Hartley and Andrew Zisserman

• Course Notes
• http://cvg.ethz.ch/teaching/compvis/tutorial.pdf

Institute of Visual Computing

Camera Calibration

 X

t R K x PX x |  

3D points 2D points

• Intrinsic parameters
• K
• Radial distortion coefficients

Institute of Visual Computing

Camera Calibration

Y X Z

• Use your own camera
• Build your own calibration object
• Print checkerboard patterns
• Stich to two orthogonal planes

Institute of Visual Computing

Camera Calibration

• 4 Tasks:
• Data normalization
• Direct Linear Transform (DLT)
• Gold Standard algorithm
• Bouguet‘s Calibration Toolbox
• Use the same settings for all tasks!
• Good reference:

Multiple View Geometry in computer vision (Richard Hartley & Andrew Zisserman)

Institute of Visual Computing

Data normalization

• Shift the centroid of the points to the origin
• Scale the points so that average distance to the
• rigin is and , respectively.
• Determine

using normalized points.

• Determine

1 3 3 3 1 2 2

1 1

 

                       

z D y D x D y D x D

c s c s c s c s c s U T

3

U P T P ˆ

1 



2

P ˆ

Institute of Visual Computing

Direct Linear Transform (DLT)

                                                  

4 , 3 3 , 3 2 , 1 1 , 1 3 2 1

1 1 P P P P y X y X y X y X X X x X x X x X x X X X X y X w X x X w

i iz i iy i ix i iz iy ix i iz i iy i ix i iz iy ix T i i T i i T T i i T T i i

 P P P AP

Institute of Visual Computing

Direct Linear Transform (DLT)

• Singular Value Decomposition

A

12 2n

U

2n 2n

S

12 2n

V‘

12 12

=

Institute of Visual Computing

Direct Linear Transform (DLT)

• Singular Value Decomposition

A

12 2n

U

2n 2n

S

12 2n

V‘

12 12

= =

               

4 , 3 3 , 3 2 , 1 1 , 1

P P P P           

4 , 3 3 , 3 2 , 3 1 , 3 4 , 2 3 , 2 2 , 2 1 , 2 4 , 1 3 , 1 2 , 1 1 , 1

P P P P P P P P P P P P

Institute of Visual Computing

Camera Matrix Decomposition (K and R)

• K is upper triangular
• R is orthonormal
• QR decomposition A = QR
• Q is orthogonal
• R is upper triangular

   

KRC KR t R K P    | |

Institute of Visual Computing

Camera Matrix Decomposition (K and R)

• Run QR decomposition on the inverse of the left

3x3 part of P

• Invert both result matrices to get K and R

   

1 1 1

| |

   

    K R M KR M KRC KR t R K P

Institute of Visual Computing

Camera Matrix Decomposition (C)

• The camera center is the point for which
• This is the right null vector of P ( SVD)

 PC

Institute of Visual Computing

Gold Standard Algorithm

• Normalize data
• Run DLT to get initial values
• Compute optimal by minimizing the sum of

squared reprojection errors

• Denormalize

P ˆ P ˆ

2 ˆ

) ˆ ˆ ˆ d( min 

 N 1 i i i

X P , x

P

Institute of Visual Computing

Minimization in MATLAB

• Fminsearch(...)
• See code framework
• Lsqnonlin(...)
• nonlinear least-squares
• Vectorize your parameters

Institute of Visual Computing

Bouguet‘s Calibration Toolbox

• Download and install the toolbox:

(http://www.vision.caltech.edu/bouguetj/calib_doc/index.html)

• Go through the tutorial and learn how to

calibrate a camera with that toolbox

• Print your own calibration pattern (available on

the website)

• Use the toolbox for calibration and compare the

result with the results of your own calibration algorithm

Institute of Visual Computing

Hand-in

• Source code
• Matlab .mat file with hand-clicked 3D-2D

correspondences

• Image used for calibration
• Visualize hand-clicked points and reprojected 3D

points

• Discuss and compare values of calibration obtained

for all methods

• Discuss average reprojection error of all methods.

Institute of Visual Computing

Some hints

• Work with normalized homogeneous coordinates always.

Camera calibration K should respect this convention.

• Check that the obtained orientation R correspond to the

expected world value, otherwise K will have negative values in its diagonal.

• When reprojecting points use average of reprojection error

for comparison and remember to normalize reprojected coordinates to w = 1.

• Remember to use the same camera with the same settings

for all tasks!

Institute of Visual Computing

Hand-in

• Example reprojection of the the 3D points